Page 200 - Mechanics of Asphalt Microstructure and Micromechanics

192 Ch a p t e r S i x

problems by using the elasticity theory. Quite a few examples were presented in Find-
ley et al. (1989).
Nevertheless, the above principle is not valid if the boundary conditions such as the
contact areas are changing with time.

6.3.11 Relationship between Creep Compliance and Relaxation Modulus
Apply the Laplace transform to the viscoelastic stress-strain relationships:
ε() = ∫ t ( C t − τ) ∂ σ τ d σ()t = ∫ t ( E t − τ) ε ∂ τ d (6-137)
t
0 τ ∂ 0 τ ∂
ε sC s σ
ˆ() s =
ˆ
() ˆ () s
(6-138)
σ sE s ε
ˆ
ˆ () s =
() ˆ() s

σ = ˆ () s = 1
ˆ () s
ε sE sC ˆ () s (6-139)
ˆ() s

or 1
ˆ
() () =
Cs E ˆ s 2 (6-140)
s
Apply the inverse Laplace transform to 6-140:
t
∫ Ct ( − ξ E ) ( )ξ d =ξ t (6-141)
0
or
t
∫ Et ( − ξ C ) ( )ξ d =ξ t
0
=
E() C() 1
0
0
∞
=
∞
E() C() 1 (6-142)

Generalization of the integral representation to 3D:
ε t() = ∫ C ( − ξ σ ξ ξ (6-143)
t
)
d
)
(
ij ijkl kl
Or in the deviatoric stress-strain relationship, and the bulk stress relationship:
ξ
t ∂ d ()
(
st() = 2 ∫ G t − ξ ) ij dξ (6-144)
ij ∂ξ
0
t ∂ εξ()
σ t() = 3 ∫ K t − ξ) ij ξ d (6-145)
(
ij ξ ∂
0
G(t) is the stress relaxation modulus in shear.
The linear stress-strain relationship in elasticity σ = λε δ + 2 G ε can be extended
ij kk ij ij
to the viscoelasticity as:
t ∂ εξ() t ∂ εξ()
σ t() = δ λε t() − ∫ ψ t − ξ) kk ξ d ]+ 2Gε () − ∫ ψ ( t ξ) ij dξ (6-146)
−
t
(
[
G
ij ij kk 1 ξ ∂ ij 2 ξ ∂
0 0